Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(6,10)(7,13)(8,15)(18,20)(21,22,25,26)(23,27,29,34)(24,31,33,28)(30,36,32,35)(38,39)(40,47)(44,46)(48,49);(6,10)(7,13)(12,17)(14,19)(22,26)(23,29)(28,31)(30,32)(37,38,42,39)(40,45,47,43)(41,44,50,46)(48,52,49,51);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,22)(23,27)(24,28)(25,26)(29,34)(30,35)(31,33)(32,36)(37,39,42,38)(40,45,47,43)(41,46,50,44)(48,52,49,51);(1,2)(3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,23,25,29)(22,27,26,34)(24,32,33,30)(28,36,31,35)(37,40,42,47)(38,43,39,45)(41,49,50,48)(44,52,46,51);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24,25,33)(22,28,26,31)(23,30,29,32)(27,35,34,36)(37,41,42,50)(38,44,39,46)(40,48,47,49)(43,51,45,52)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=fh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4775452);